Sharp Hardy inequalities in the half space with trace remainder term

In this paper we deal with a class of inequalities which interpolate the Kato's inequality and the Hardy's inequality in the half space. Starting from the classical Hardy's inequality in the half space $\rnpiu =\R^{n-1}\times(0,\infty)$, we show that, if we replace the optimal constant $\frac{(n-2)^2}{4}$ with a smaller one $\frac{(\beta-2)^2}{4}$, $2\le \beta <n$, then we can add an extra trace-term equals to that one that appears in the Kato's inequality. The constant in the trace remainder term is optimal and it tends to zero when $\beta$ goes to $n$, while it is equal to the optimal constant in the Kato's inequality when $\beta=2$.